# Facing Missing Observations in Data—A New Approach for Estimating Strength of Earthquakes on the Pacific Coast of Southern Mexico Using Random Censoring

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Methodology

#### 2.2.1. The GEV Distribution

#### 2.2.2. Estimation of the Parameters of the GEV under Random Censoring

#### 2.2.3. Bayesian Implementation

#### 2.2.4. Simulation Study

#### 2.3. Data Collection

#### 2.4. Data Analysis

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EVT | Extreme Value Theory |

MCMC | Markov Chain Monte-Carlo |

SASMEX | Sistema de Alerta Sísmica de Mexico |

SAS | Sistema de Alerta Sísmica de la Ciudad de Mexico |

SASO | Sistema de Alerta Sísmica de Oaxaca |

CIRES | Centro de Instrumentación y Registro Sísmico |

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**Figure 2.**(

**a**) Real functions and (

**b**,

**c**) functions obtained by fitting the parameters of a non-stationary GEV model with ${P}_{2}=20$ and ${P}_{2}=80$ knots, respectively, to simulated data with a sample size of $n=500$.

**Figure 3.**Random censoring. X, event; M, maxima; U, censored observation. The blue dotted line shows the limits of the blocks.

**Figure 4.**Estimated spatial smoothing of the location parameter for the years: (

**a**) 1995; (

**b**) 2000; (

**c**) 2005; (

**d**) 2010; (

**e**) 2015; (

**f**) 2018.

**Figure 5.**Estimated temporal smoothing of the location parameter of earthquake maxima at six geographical locations on the Pacific coast of southern Mexico.

**Table 1.**Descriptive summary information on the Maxima of earthquakes on the Pacific coast of southern Mexico.

Block ID | Long $\left(\mathit{W}\right)$ | Lat $\left(\mathit{N}\right)$ | Min. | 1st Qu. | Median | Mean | 3rd Qu. | Max. |
---|---|---|---|---|---|---|---|---|

1 | −91.5${}^{\circ}$ | 14${}^{\circ}$ | 6.1 | 6.2 | 6.8 | 6.7 | 7.3 | 7.3 |

2 | −91.5${}^{\circ}$ | 15${}^{\circ}$ | 6.9 | 6.9 | 7 | 7 | 7 | 7 |

3 | −91.5${}^{\circ}$ | 16${}^{\circ}$ | 4.7 | 4.9 | 5.1 | 5.1 | 5.2 | 5.4 |

4 | −94.5${}^{\circ}$ | 15${}^{\circ}$ | 4.2 | 5 | 5.3 | 5.5 | 5.8 | 8.2 |

5 | −94.5${}^{\circ}$ | 16${}^{\circ}$ | 3 | 4.1 | 4.3 | 4.4 | 4.7 | 6.6 |

6 | −94.5${}^{\circ}$ | 17${}^{\circ}$ | 3.8 | 4.2 | 4.4 | 4.5 | 4.7 | 6.7 |

7 | −94.5${}^{\circ}$ | 18${}^{\circ}$ | 4 | 4.4 | 4.6 | 4.8 | 4.9 | 6.4 |

8 | −94.5${}^{\circ}$ | 19${}^{\circ}$ | 5.5 | 5.5 | 5.5 | 5.5 | 5.5 | 5.5 |

9 | −97.5${}^{\circ}$ | 15${}^{\circ}$ | 4.1 | 4.2 | 4.6 | 4.6 | 5 | 5.1 |

10 | −97.5${}^{\circ}$ | 16${}^{\circ}$ | 2.9 | 3.9 | 4 | 4.2 | 4.3 | 7.4 |

11 | −97.5${}^{\circ}$ | 17${}^{\circ}$ | 2.2 | 3.8 | 4 | 4.1 | 4.2 | 5.8 |

12 | −97.5${}^{\circ}$ | 18${}^{\circ}$ | 3.8 | 4.1 | 4.4 | 4.7 | 5.3 | 7.1 |

13 | −97.5${}^{\circ}$ | 19${}^{\circ}$ | 4.8 | 4.8 | 4.8 | 4.8 | 4.8 | 4.8 |

14 | −100.5${}^{\circ}$ | 16${}^{\circ}$ | 3.6 | 4 | 4.3 | 4.3 | 4.6 | 5.2 |

15 | −100.5${}^{\circ}$ | 17${}^{\circ}$ | 3.1 | 3.9 | 4.1 | 4.2 | 4.4 | 7.2 |

16 | −100.5${}^{\circ}$ | 18${}^{\circ}$ | 3.5 | 4 | 4.1 | 4.4 | 4.4 | 6.5 |

17 | −100.5${}^{\circ}$ | 19${}^{\circ}$ | 4.2 | 4.3 | 4.4 | 4.3 | 4.4 | 4.4 |

18 | −103.5${}^{\circ}$ | 17${}^{\circ}$ | 5.1 | 5.1 | 5.1 | 5.1 | 5.1 | 5.1 |

19 | −103.5${}^{\circ}$ | 18${}^{\circ}$ | 3.7 | 4 | 4.3 | 4.5 | 4.7 | 6.5 |

20 | −103.5${}^{\circ}$ | 19${}^{\circ}$ | 3.7 | 4 | 4.1 | 4.3 | 4.4 | 5.9 |

21 | −106.5${}^{\circ}$ | 19${}^{\circ}$ | 4.1 | 4.2 | 4.2 | 4.8 | 5.3 | 6.3 |

22 | −106.5${}^{\circ}$ | 20${}^{\circ}$ | 3.8 | 4 | 4.1 | 4.2 | 4.3 | 4.6 |

**Table 2.**Estimates and 95% credible intervals of the non-stationary GEV model for the maxima of earthquakes.

Coef. | Mode | Mean | 95% CI |
---|---|---|---|

${\beta}_{\left(1\right)0}$ | 0.8136 | 0.7004 | (0.4677, 0.8079) |

$\sigma $ | 0.5710 | 1.1110 | (1.0855, 1.1697) |

$\kappa $ | −0.5900 | −0.5930 | (−0.6014, −0.5900) |

${\sigma}_{1}^{2}$ | 26.1363 | 83.3487 | (26.2751, 810.0938) |

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**MDPI and ACS Style**

Aguirre-Salado, A.I.; Vaquera-Huerta, H.; Aguirre-Salado, C.A.; Jiménez-Hernández, J.d.C.; Barragán, F.; Guzmán-Martínez, M.
Facing Missing Observations in Data—A New Approach for Estimating Strength of Earthquakes on the Pacific Coast of Southern Mexico Using Random Censoring. *Appl. Sci.* **2019**, *9*, 2863.
https://doi.org/10.3390/app9142863

**AMA Style**

Aguirre-Salado AI, Vaquera-Huerta H, Aguirre-Salado CA, Jiménez-Hernández JdC, Barragán F, Guzmán-Martínez M.
Facing Missing Observations in Data—A New Approach for Estimating Strength of Earthquakes on the Pacific Coast of Southern Mexico Using Random Censoring. *Applied Sciences*. 2019; 9(14):2863.
https://doi.org/10.3390/app9142863

**Chicago/Turabian Style**

Aguirre-Salado, Alejandro Ivan, Humberto Vaquera-Huerta, Carlos Arturo Aguirre-Salado, José del Carmen Jiménez-Hernández, Franco Barragán, and María Guzmán-Martínez.
2019. "Facing Missing Observations in Data—A New Approach for Estimating Strength of Earthquakes on the Pacific Coast of Southern Mexico Using Random Censoring" *Applied Sciences* 9, no. 14: 2863.
https://doi.org/10.3390/app9142863